1. Chapter 5 Integrals 5.2 The Definite Integral
In this handout:
Riemann sum
Definition of a definite integral
Properties of the definite integral

2. Riemann Sum
An ordered collection P=(x0,x1,…,xn) of points of a closed interval I = [a,b] satisfying a = x0 < x1 < …< xn-1 < xn = b is a partition of the interval [a,b] into subintervals Ik=[xk-1,xk].
Let Δxk = xk-xk-1
For a partition P=(x0,x1,…,xn), let |P| = max{Δxk, k=1,…,n}. The quantity |P| is the length of the longest subinterval Ik of the partition P.
|P|
a = x0 x1 x2
xn-1
xn =b
Choose a sample point xi* in the subinterval [xk-1,xk].
A Riemann sum associated with a partition P and a function f is defined as:

3. Definition of a Definite Integral
If f is a function defined on [a, b], the definite integral of f from a to b is the number
provided that this limit exists. If it does exist, we say that f is integrable on [a, b].
upper limit of integration
Integration
Symbol
variable of integration
(dummy variable)
integrand
lower limit of integration
Note that the integral does not depend on the choice of variable.

4. Existence of a Definite Integral
Theorem: If f is continuous on [a, b],
or if f has only a finite number of jump discontinuities, then f is integrable on [a, b].
If f is integrable on [a, b], then in calculating the value of an integral we are free to choose the partitions and sample points to simplify the calculations.
It is often convenient to take a regular partition;
that is, all the subintervals have the same length Δx.

5. Properties of the Integral
Reversing the limits changes the sign.
If the upper and lower limits are equal, then the integral is zero.
Constant multiples can be moved outside.
where c is any constant
Integrals can be added (or subtracted).
Intervals can be added (or subtracted.)

6. Comparison Properties of the Integral
If f(x) ≥ 0 for a ≤ x ≤ b, then
If f(x) ≥ g(x) for a ≤ x ≤ b, then
If m ≤ f(x) ≤ M for a ≤ x ≤ b, then